For large, parameterized multiscale problems, RB methods need to be combined with numerical multiscale methods, as otherwise the solution of the FOM for a single snapshot parameter might already be computationally infeasible.
One possible approach is to speed up the solution of the individual cell problems using RB techniques, as is done in [1]}
in the context of numerical homogenization, in [2]}, [3]}, [4]}, [5]}, [6]} for the HMM, in [7]}, [8]}, [9]} for the MsFEM, or in [10]} for the LOD.
This approach is applicable both for problems, where the fine-scale data variation over the domain is parameterized, resulting in a single ROM for all cell problems [2]}, [3]}, [4]}, [5]}, [6]}, [1]}, [8]}, or for general parameterized problems, where for each individual cell problem a dedicated ROM is built [10]}, [9]}.
Some recent works have considered the case where the multiscale data varies only in some cells, e.g., caused by perturbations, by a sequence of modification over time or by other parameterizations, such that individual fine-scale solutions can be reused [20]}, [21]}, [22]} or a ROM can be built in case of parameterized perturbations [23]}.
